Introduction to Functions
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Questions and Answers

What is the inverse operation of addition?

  • Multiplication
  • Subtraction (correct)
  • Exponentiation
  • Division
  • The range of a function can also refer to its outputs.

    True

    What is the domain of a function related to?

    Input values

    The inverse of multiplication is __________.

    <p>division</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Domain = Set of all possible input values for a function Range = Set of all possible output values for a function Function = A relationship where each input has exactly one output Inverse = An operation that reverses the effect of another operation</p> Signup and view all the answers

    If a function f(x) has a domain of all real numbers, which of the following is true?

    <p>It can have any type of input.</p> Signup and view all the answers

    All functions must be linear.

    <p>False</p> Signup and view all the answers

    What must an element of a domain have for a function to be defined?

    <p>Unique output</p> Signup and view all the answers

    For a function to be considered valid, it is necessary that each input must have __________ values.

    <p>exactly one output</p> Signup and view all the answers

    Which of the following statements is true regarding elements of a domain?

    <p>Each element must have a unique output.</p> Signup and view all the answers

    What condition must a function meet to be considered one-to-one?

    <p>Each output must correspond to exactly one input.</p> Signup and view all the answers

    A function that is not one-to-one may still have an inverse.

    <p>False</p> Signup and view all the answers

    What does the notation (5,4) represent in the context of coordinates?

    <p>(5, 4) is a point in a two-dimensional space.</p> Signup and view all the answers

    A function that is _____ must have a unique output for every input.

    <p>one-to-one</p> Signup and view all the answers

    Which of the following represents a valid function output for the input (5,10)?

    <p>(5,10)</p> Signup and view all the answers

    A function can have the same output for different inputs.

    <p>True</p> Signup and view all the answers

    What happens if a function has multiple outputs for the same input?

    <p>It is not a valid function.</p> Signup and view all the answers

    The range of a function is determined by its _____ values.

    <p>output</p> Signup and view all the answers

    What is the purpose of the vertical line test?

    <p>To determine if a graph represents a function</p> Signup and view all the answers

    Signup and view all the answers

    Study Notes

    Introduction to Functions

    • A function is a relationship between inputs where each input is related to exactly one output.
    • Functions have a domain (set of inputs) and a codomain/range (set of possible outputs).
    • Functions are typically denoted as f(x), where x represents the input.
    • Examples of functions include: f(x) = sin x, f(x) = x² + 3, f(x) = 1/x, and f(x) = 2x + 3.

    Types of Functions

    • One-to-one function (Injective Function): Each input maps to a unique output, and no two inputs map to the same output.

    • Many-to-one function: Multiple inputs can map to the same output.

    • Onto function (Surjective Function): Every element in the codomain is mapped to by at least one element in the domain.

    • Bijective Function: A function that is both one-to-one and onto. Invertible functions are bijective.

    • Other function types mentioned are: Linear, Quadratic, Rational, Algebraic, Cubic, Modulus, Signum, Greatest Integer, Fractional Part, Even and Odd, and Periodic.

    Invertible Functions

    • The inverse of a function f, denoted as f⁻¹(x), reverses the relationship, mapping outputs back to inputs.
    • A function is invertible if and only if it is bijective (one-to-one and onto).
    • For a function to have an inverse, each output must correspond to a unique input.
    • The inverse of adding is subtracting, and the inverse of multiplying is dividing.

    Graphical Representation

    • The domain of a function is the set of inputs, and the range is the set of outputs.
    • The domain of f⁻¹(x) is the range of f(x), and the range of f⁻¹(x) is the domain of f(x).
    • Visual representation aids understanding of function relationships and inverses.

    Reflexive Property of Invertible Functions

    • If a point (a, b) is on the graph of a function f, then the point (b, a) must be on the graph of the inverse function f⁻¹.
    • This property demonstrates the reflection of the graph of a function over the line y = x when finding the inverse.

    Conditions for Invertibility

    • A function is invertible if it's both one-to-one and onto over its entire domain.
    • A function may be one-to-one in certain parts of its domain but not over the entire domain.

    Horizontal Line Test

    • Used to determine if a function is invertible (one-to-one).
    • If any horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse.

    Demonstration of an Invertible Function

    • Graphing y = 2x and y = x/2, illustrates the relationship between a function and its inverse.
    • The inverse graph is a reflection of the original graph across the line y = x.

    Properties of the Inverse Graph

    • The inverse of a function is found by exchanging x and y coordinates in its graph.
    • Graph of the inverse of f is a reflection across the y = x line.
    • Example in page 9 details coordinates shifting in this relation.

    Conclusion regarding invertibility

    • Functions must be one-to-one to have an inverse.
    • Exchanging input and output rows is how to find the inverse of tabular functions.
    • Many "toolkit functions" have inverses.
    • Graphically, an inverse is a reflection across the line y = x.

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    Function Notes PDF

    Description

    This quiz covers the fundamentals of functions, including their definitions, types, and notations. Students will explore one-to-one, many-to-one, onto, and bijective functions. Enhance your understanding of how functions relate inputs to outputs with practical examples.

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